The original poster gave me some points! ! !
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The history of the development of non-Euclidean geometry
1. The question 1 was raised
The development of non-Euclidean geometry originated from Euclid's "Elements of Geometry" by the ancient Greek mathematician more than 2,000 years ago. 》. Postulate 5 was proposed by Euclid himself. Its content is: "If a straight line intersects two straight lines, and if the sum of two interior angles on the same side is less than two right angles, then the two straight lines must intersect on that side after being infinitely extended. This postulate has aroused widespread discussion because it is not as concise as other axioms and postulates. Euclid himself was not satisfied with this postulate. He only used it after proving all theorems that did not require the parallel postulate. , it is suspected that it may not be an independent postulate, and may be replaced by other postulates or axioms. For more than 2,000 years, from the ancient Greek times to the 19th century, mathematicians have always been worried about this postulate and have worked tirelessly to solve this problem. Mathematicians mainly proceed along two research paths: one path is to find a more self-evident proposition to replace the parallel postulate; the other path is to try to deduce the parallel postulate from the other 9 axioms and postulates, and found it along the first path. The simplest expression of the fifth postulate was given by the Scottish mathematician Playfair (J. Playfair 1748-1819) in 1795: "Through a point outside the straight line, there is and is only one straight line parallel to the original straight line." This is our middle school textbook today. The parallel axiom used in this paper, but in fact it was stated by the ancient Greek mathematician Proclus in the 5th century AD. The problem is, however, that all these alternative postulates are no more acceptable or more "natural" than the original fifth postulate. The first major attempt in history to prove the Fifth Postulate was made by the ancient Greek astronomer Ptolemy (about 150 AD). Later Proclus pointed out that Ptolemy's "proof" inadvertently assumed the past. Only a straight line parallel to the known straight line can be drawn from a point outside the straight line. This is the Playfield postulate mentioned above
Solution to the 1.2 problem
1.2.1 The bud of non-Euclidean geometry
The work of demonstrating the fifth postulate along the second path made breakthrough progress in the 18th century. First, the Italian Sacchairi (1667-1733) proposed to use the reductio method to prove the fifth postulate. Sacchairi started from the quadrilateral ABCD. If angle A and angle A are right angles, and AC=BD, it is easy to prove that angle C is equal to angle D. ． In this way, the fifth postulate is equivalent to the assertion that angle C and angle D are right angles. Sacheri put forward two other hypotheses: (1) Obtuse angle hypothesis: Angle C and angle D are both obtuse angles; (2) Acute angle hypothesis: Angle C and angle D are both acute angles. Finally, under the assumption of acute angles, Sacheri derived a series of results. Because they were contrary to empirical knowledge, he gave up the final conclusion. However, objectively it provides a very valuable way of thinking for the creation of non-Euclidean geometry and opens up a new way that is different from the predecessors. Later, the Swiss mathematician Lambert (Lambetr1728-1777) did work similar to Saccheri. He also examined a class of quadrilaterals in which three angles were right angles and the fifth angle had three possibilities: right angle, obtuse angle, and acute angle. He also obtained under the assumption of acute angles that "the area of ??a triangle depends on the sum of its interior angles; the area of ??a triangle is proportional to the difference between the straight angle and the sum of the interior angles. He believed that as long as a set of assumptions does not contradict each other, it provides a geometric possibility. The famous The French mathematician Legendre (A.M. Legendar 1752-1833) was also very concerned about the parallel postulate. He obtained an important theorem: "The sum of the interior angles of a triangle cannot be greater than two right angles." This indicates that there may be a new kind of Geometry, in the early 19th century, the German Schweikart (1780-1859) made this idea more clear. Through his research on "star geometry", he pointed out: "There are two types of geometry: geometry in the narrow sense (Euclidean geometry) ) Star geometry, in the latter one, a characteristic of triangles is that the sum of the interior angles of a triangle is not equal to two right angles."
1.2,2 The birth of non-Euclidean geometry
Some of the mathematicians mentioned above, especially Lambert, are pioneers of non-Euclidean geometry, but they have not formally proposed a new geometry and established its systematic theory. The famous mathematicians Gauss (Gauss1777-1855), Wave John (Bolyai 1802-1860) and Lobatchevsky (Lobatchevsky 1793-1856) did this and became the founders of non-Euclidean geometry. Gauss was the first person to point out that Euclid's fifth postulate is independent of other postulates. In 1792, he already had an idea to establish a logical geometry in which Euclid's fifth postulate did not hold. In 1794, Gauss discovered that in his geometry, the area of ??a quadrilateral was proportional to the two square angles and the quadrilateral. The difference between the sum of the interior angles, and thus derived that the area of ??a triangle does not exceed a constant, no matter how far apart its vertices are. Later he further developed his new geometry, which he called non-Euclidean geometry. There is no contradiction, and it is real and applicable. For this reason, he also measured the interior angles of the triangle formed by 3 mountain peaks. He believed that the deficiency of the sum of the interior angles can only be revealed in very large triangles. However, his measurement failed due to the error of the instrument. Unfortunately, Gauss did not have any writings on non-Euclidean geometry during his lifetime. People learned about his non-Euclidean geometry from his correspondence with his friends after his death. Research results and opinions on Euclidean geometry.
2.
. Enlightenment from the development history of non-Euclidean geometry
The birth of non-Euclidean geometry is a major innovative step in mathematics since the Greek era. Here we will describe the important significance of this history along the historical development process of things. M. When evaluating this period of history, M. Klein said: "The history of non-Euclidean geometry shows in a surprising way how much mathematicians are influenced by the spirit of their times. At that time, Saccheri rejected Euclidean geometry. strange theorem, and concluded that Euclidean geometry was the only correct one. But a hundred years later, Gauss, Lobachevsky and Boyo accepted the new geometry with confidence."
2.1 The subject of mathematics itself
2.1.1 The relative independence of the development of mathematics
The non-Euclidean geometry system established through logical deduction is The development of mathematics provides a model that allows people to clearly see that mathematics can have its own logical system and thus develop independently. The relative independence of the development of mathematics is highlighted as follows: the development of mathematical theory is often advanced, it can be carried out independently of the physical world, it can be ahead of social practice, and react on social practice, promoting the development of mathematics and even the entire science. Before the 19th century, mathematics was always closely integrated with applied mathematics, that is, mathematics could not develop independently from practical subjects. The ultimate purpose of studying mathematics was to solve practical problems. However, for the first time, non-Euclidean geometry made the development of mathematics ahead of practical sciences. , beyond people's experience, non-Euclidean geometry creates a new world for mathematics: human beings can use their own thinking to think freely according to the logical requirements of mathematics. Mathematics was then considered to be those arbitrary structures that did not arise directly or indirectly from the need to study nature. This view was gradually understood by people, which resulted in the split between pure mathematics and applied mathematics today LlJ.
2.1.2 The essence of mathematics lies in its full freedom
The creation of non-Euclidean geometry has brought to light differences that people have always been aware of but have not clearly understood. The difference between mathematical space and physical space. Mathematicians create m-geometry theory and then determine their view of space based on it. This view of space and nature based on mathematical theory generally does not deny the existence of the objective world. It only emphasizes the following facts: People The series of conclusions obtained from the judgment about space are purely his own creation. The reality of the material world and the theory of this reality are always two different things. Because of this, human beings' cognitive activities of exploring knowledge and establishing theories will never end. The creation of non-Euclidean geometry made people realize that mathematics is a creation of the human spirit, rather than a direct copy of objective reality. This has made mathematics gain great insights, and at the same time, mathematics has lost its certainty about reality. ． Mathematics is freed from nature and science and continues its own journey. In this regard, M. Klein said: "This stage in the history of mathematics freed mathematics from its close connection with reality and separated mathematics itself from science, just as science separated from philosophy, philosophy from religion, and religion Separate from animism and superstition. Now we can use the words of George Cantor: "The essence of mathematics lies in its complete freedom".
2.1.3 Update of geometric concepts
The emergence of non-Euclidean geometry broke the dominance of Euclidean geometry and updated the concept of geometry. Traditional Euclidean geometry believes that space is unique, but the emergence of non-Euclidean geometry breaks this concept and prompts people to conduct in-depth discussions on the basic issues of Euclidean geometry and even the entire geometry.
2.2 Cultural and educational aspects
2.2.1 Non-Euclidean geometry is the product of the human spirit who dare to challenge tradition and devote themselves to science. Gauss, Boyo, and Robard Chevsky discovered non-Euclidean geometry almost at the same time, but the three people had different attitudes towards the new geometry. Gauss was aware of the existence of new geometry very early, but he did not announce his new ideas to the world. He was influenced by Kant's idealism and did not dare to challenge the 2000-year-old Euclidean geometry in the traditional geometry circle. As a result, the birth of non-Euclidean geometry was delayed. Boyo devoted himself to the study of the parallel postulate and finally discovered new geometry. There is also a story. When Gauss decided to keep his discovery secret, Boyette was eager to make his research public through Gauss's evaluation. However, Gauss wrote back to his father F. Boyette said: " Praising him is equivalent to praising myself. The content of the entire article, the ideas adopted by your son and the results obtained coincide with my thinking 30 to 35 years ago." Boyo was deeply disappointed with Gauss's answer. .
Thinking that Gauss wanted to plagiarize his own work, especially after the publication of Lobachevsky's work on non-Euclidean geometry, he decided not to publish any more papers.
After Lobachevsky disclosed his new geometric ideas in 1826, he was not understood and praised by his contemporaries. Instead, he was ridiculed and attacked, "But no force can shake Lobachevsky." Fsky's confidence was like a lighthouse standing in the sea. The impact of the stormy waves fully demonstrated his resolute will. He always struggled for new ideas throughout his life. When he was blind, he also dictated the completion of "" "Pan Geometry".
The process of three people's discovery of new geometry enlightens us: only by breaking through the superstition of tradition and authority can we give full play to the creativity of science; only by not being afraid of hardships and having the courage to devote ourselves to science , in order to pursue and defend the truth that transcends the times. It is generally believed that Gauss, Boyo, and Lobachevsky discovered the new geometry at the same time. This is people's justice to history, but people prefer to call the new geometry Loche geometry. This is people's high praise for Lobachevsky's dedication to science
The spirit of 2, 2, 2 non-Euclidean geometry encourages people to establish a tolerant and inclusive attitude. Product
The creation of non-Euclidean geometry liberated human thought, new insights and new perspectives continued to emerge, "mathematics appears as the free creation of human thought" 5]. The development of mathematics made Cantor sincerely say. : "The essence of mathematics lies in its freedom." This intellectually active and democratic artistic atmosphere has enabled mathematics to develop at an unprecedented speed. The tortuous creation process of non-Euclidean geometry and the development of mathematics it has brought about have made people realize To the importance of free creation and contention of a hundred schools of thought to scientific development, it encourages people to establish the spirit and virtue of tolerance and tolerance [6.
2.3 Philosophical aspects
2.3. 1 Changes in epistemology
French philosopher and mathematician Henri Poincare said 7: The discovery of non-Euclidean geometry is the root of a revolution in epistemology. Simply put, one can say that this discovery. It has successfully broken through the dilemma required by traditional logic and binding any theory: that is, scientific principles are either necessary truths (transcendental synthetic logical conclusions); or asserted truths (facts of sensory observation). He pointed out: Principles may be simple arbitrary agreements, but these agreements are by no means unrelated to our minds and nature. They can only exist through the tacit understanding of all people, and they are closely dependent on the environment in which we live. Actual external conditions. In fact, it is precisely because of this that when exploring unknown or currently insensible things, we can rely on our understanding of nature to make some kind of "tacit agreement" in the field of philosophy. This is the beginning of understanding all things. And the basis. In addition, we should give up the judgment of either/or in theoretical evaluation. Einstein said 8]: This judgment of these critics and mathematicians is undoubtedly incorrect. After the creation of right and wrong Euclidean geometry, it had the most direct impact on the establishment of ideas and theories, especially on epistemology. Further modern theoretical and technological progress are inseparable from its intrinsic influence, such as the emergence of the "theory of relativity", especially the theory of relativity. It is a further understanding of space and time. The establishment and development of set theory, modern analysis foundation, mathematical logic, quantum mechanics and other disciplines can be seen as a direct result of non-Euclidean geometry. The shock caused by the creation of non-Euclidean geometry has not subsided to this day.
2.3.2 Breaking the traditional way of thinking of mankind
The primary basis for analyzing and evaluating a theory should be to see whether it has "compatibility", that is, whether it has Contradictory conclusions may be drawn if a theory does not yet "justify itself."